Efficient protein synthesis depends on the availability of charged tRNA molecules. With 61 different codons, shifting the balance among the tRNA abundances can lead to large changes in the protein synthesis rate. Previous theoretical work has asked about the optimization of these abundances, and there is some evidence that regulatory mechanisms bring cells close to this optimum, on average. We formulate the tradeoff between the precision of control and the efficiency of synthesis, asking for the maximum entropy distribution of tRNA abundances consistent with a desired mean rate of protein synthesis. Our analysis, using data from E. coli, indicates that reasonable synthesis rates are consistent only with rather low entropies, so that the cell's regulatory mechanisms must encode a large amount of information about the "correct" tRNA abundances.